Visualizing the phase

There are different tricks to visualize the phase. The one you hint at in your answer, amounts to taking the real (or imaginary) part of the wavefunction. For instance, ${rm Re}[e^{i (k x + phi_0)}]=cos(kx + phi_0)$, which is a sinusoid. The phase determines the location and distance between the crests and troughs. Another trick is to use color -- if you google "complex plane color" you will find many pictures and explanations of this way of looking at this. Yet another visualization is to imagine a little clock face at every point in space, and the position of the minute hand on the clock tracks the phase of the wavefunction. Keep in mind that all of these are simply different representations and are not "what the phase really is"; it's useful to know understand several different ways to visualize the phase, and use the one that is the most convenient or insightful in a particular instance.

Note added, thanks to jgerber's comment: There are some very nice visulations here: vqm.uni-graz.at

Consequence of the phase: interference

The fact that complex probability amplitudes, rather than real-valued probabilities, add in quantum mechanics leads to interference effects. Here we can think about the classic double slit experiment.

Classically, the probability that the particle goes through one slit simply adds to the probability that the particle goes through the other slit. Since probabilities are positive, there are no points with zero probability on the screen which catches the particles.

On the other hand, in quantum mechanics, there are points on the screen where the probability amplitude to go through slit 1 is $a$, and the probability amplitude to go through slit 2 differs by a phase, $e^{i pi}a=-a$, so the sum of the probability amplitudes is zero and there is zero probability to find the particle at this location. At other locations on the screen, the relative phase will be +1, and the probability amplitudes will add constructively, leading to a "large" probability to find the particle at these locations. In general, the relative phase of the probability amplitude in passing through the two slits is what determines the shape of the interference pattern.

Variations of the phase encode physical information

In general, the phase of the wavefunction $Psi$ appears in the definition of the "probability current", $vec{j}propto i (Psi^star nabla Psi - Psi nabla Psi^star)$. A state which is purely real has a vanishing probability current, which is easy to see from the definition. Thus having a non-zero phase allows the state to have a non-zero current, and thus allows the state to change.

We can give this a more direct meaning in some special cases, where the phase of the wavefunction (or more exactly the derivative of the phase with respect to some parameter) encodes an observable quantity. (This is related to the idea of action angle variables in classical mechanics).

This is exactly true when eigenvalue problems for the wave function $Psi$ of the following form appear begin{equation} i frac{partial Psi}{partial z} = lambda Psi end{equation} where $z$ is some parameter and $lambda$ is an associated eigenvalue. Note that if we write $Psi=A e^{i phi}$, where $A$ is a constant, then the above equation can be written begin{equation} -frac{partial phi}{partial z} = lambda end{equation} If $i partial/partial z$ is an operator, and $lambda$ is a possible measurable outcome of $lambda$, then the above equation says that the variation of the phase with respect to $z$ gives us the observable value $lambda$.

There are many examples of type of equation:

• If $z$ is position then $lambda$ is the momentum; for plane wave states, the derivative of the phase with respect to position is the momentum.
• If $z$ is momentum then $lambda$ is the position.
• If $z$ is time then $lambda$ is the energy.
• If $z$ is the azimuthal angle (angle in the $x-y$ plane) then $lambda$ is the component of the angular momentum parallel to the $z$ axis.

Furthermore, this kind of equation shows up in an approximate form in the WKB approximation. Then, there is a similar type of relation between, say, the derivative of the phase with respect to position, and the momentum, will approximately hold.

Having said that, don't go overboard; the interpretation is a little bit fuzzier for states which are superpositions of eigenstates, and not all observables have the form $i partial/partial z$. However, this seed of intuition is useful to keep in mind when considering more complicated observables, which have eigenfunctions more complicated than a plane wave.

TL;DR

• There are many tricks for visualizing complex numbers -- learn several.
• A non-zero phase means the wavefunction is complex, which:
  • allows interference phenomena to occur,
  • means the probability current is non-zero and therefore means the wavefunction can change in time.
• The "more wiggly" the state is with respect to position, the more momentum it has. The "wiggliness" is encoded in the phase. Similar "wiggliness" relationships exist between energy and time, and angle and angular momentum.

I love the pictures from the book of Feynman "QED: The Strange Theory of Light and Matter". There, the phase of a wavefunction traveling though the space is depicted as a clock with a moving hand: when the hand makes a full round it corresponds to $2 pi$ of the complex phase.

This can be easily used to explain the effect of interference: different paths between A and B lead to different travel times and thus to different final positions of the clock hand. Now, the hands obtained from all paths are added like vectors (clock hands look like arrows!). Similar angles add up constructively, opposite angles add up destructively.

The book contains many more examples based on this brilliant visualization idea.

The reason that the overall phase of a wavefunction is hard to gain intuition for is that it's really not real, in the sense that it's just an artifact of a particular choice of quantum formalism and doesn't appear - even purely mathematically - in other formalisms.

In particular, in the density-matrix and projective-space formalisms of quantum mechanics (which are for some purposes more useful than the state-vector formalism), you don't have the freedom to multiply the overall state by an unobservable phase factor. In these formalisms, the states $|psirangle$ and $e^{i theta} |psirangle$ are represented by literally the exact same mathematical object (a rank-one projection operator or a projective ray, respectively). And energy eigenstates don't pick up an unobservable phase under time evolution, but instead remain completely unchanged.

The reason that you're having a hard time picturing the overall phase of a quantum state is that there is truly nothing to picture.

Besides what Andrew already said, I want to add that you cannot expect to have human intuition about every physically meaningful concept or quantity or property. For example, you cannot have human intuition about the energy states of an electron in a hydrogen atom; it just falls out of the mathematics. That said, you can think of phase of a wavefunction as analogous to the phase of the electric field vector in the classical description of a light beam. It is there, and may rotate over time (as in circularly polarized light), and can be observed using detectors that are specifically designed to react differently to different polarization (such as a polarizing filter followed by a light detector).

Since you asked for non-mathematical intuition for the phase, I'd say that a classical wave is indeed a good analogy except for the fact that the global phase---as opposed to the relative phase---between two modes is completely indeterminate. Phenomena such as as interference between two modes are fully explained by visualizing them as classical waves that add up or cancel out, with the only caveat that the starting point of the "wiggling" for any of the individual modes isn't possible to determine.

Consider for example a Mach-Zehnder interferometer where only a single photon is fed in at one of the modes. Before it enters the first beam splitter, its (global) phase is unknown. However, as soon as it delocalizes after the first beam splitter, the relative phase between the two arms of the interferometer is well defined and fully accounts for which output mode the photon is likely to re-emerge from.

As a simple exercise, you could try to calculate how interaction-free measurements come about. You'll see that the classical wave picture is perfectly adequate to explain what happens between the two beam splitters, including the precise visualization of the relative phase, but it does not say anything about the global one outside the interferometer.

EDIT:

In a nutshell, if the two arms of the interferometer make up the two modes, you can start with a photon in a single mode only, i.e., the input state could be $e^{itheta}mid10rangle$ where $theta$ is the (unknowable) global phase. After the first 50/50 beam splitter, we end up with a "delocalized" photon $frac{1}{sqrt{2}}left(mid10rangle+e^{iphi}mid01rangle right)$ where $phi$ is the relative phase between the two arms. After you pass the second 50/50 beam splitter, you get a probability of $frac{1}{2} left( 1+cosphi right)$, hence the "discretized" interference pattern between the two modes modulated by $phi$. As you can see, nowhere in this discussion has the global phase entered---and that's the main difference with classical waves. (The addition of an obstacle in one of the arms, as in the interaction-free experiment, is just a bonus scenario to better understand the concept of phase.)

I understand the mathematical argument that the phase doesn't matter: the complex exponential cancels out when you calculate the probability distribution, etc.

This is not true always, as you've seen in this thread, when the particle is in superposition of two or more wavefunctions.

Maybe my confusion stems from a misunderstanding of what phase even is in quantum mechanics. When I visualize phase, I think of a sine wave and how much it has been shifted to the left or right (relative to some origin).So what information does the phase encode here? I'm clearly missing something...

This shift can be also determined for a $e^{-i(kx -omega t)}$ which is independent of the imaginary part and it can be used to calculate the group velocity and phase velocity of the wavefunction. And as many other have stated the physical meaning of the phase is not so important, even some wavefunctions are imaginary, the most important part is the physical meaning they represent.

That's because the absolute phase is not physically meaningful.

It only acquires any meaning when compared to a reference, another path that also has a phase, for instance another path from a beamsplitter, or reflected from a boundary. Then the two paths can interfere, constructively or destructively, as dictated by their phase difference.

tl;dr– You can think of how sounds can combine to form louder sounds or cancel out. Whether sounds add or cancel depends on their phase.

Phase is a more general mathematical concept. It comes up in all sorts of stuff beyond quantum-mechanics, e.g. sounds, electric-current, and radio-waves. Probably easier to understand the concept itself outside of quantum-mechanics.

Example: Phase determines if sound-waves add or cancel.

Say you set up a microphone that listens for sounds and then replays them at the same volume on a speaker. Does this result in louder sounds or noise-cancellation?

Depends on the phase!:

• If the sound-waves are lined up to peak together, the sounds'll add.

• If the sound-waves have their peaks in opposition, then they'll tend to cancel out. Which is how active noise-cancellation works.

Some observations...

Contrary to the above descriptions, sound-waves don't actually have phases in an absolute sense. We can just talk about sound-waves as though they had phases as one mode of thinking about them.

If we do choose to think of two sets of sound-waves as having phases, then the big thing is that we correctly consider their phases relative to each other. But if we choose to say that one's in the default-phase while the other's in an anti-phase or something, well, that'd be pretty arbitrary.

Sound-waves were probably the simplest example. Once you feel comfortable with phases with respect to them, the next analogy to consider might be alternating electric current (AC electricity), e.g. in three-phase electric systems.

As for math, probably best to start by learning about the Fourier transform. The gist is to redescribe things in repeating signals of different frequencies (often described as the frequency domain). In plain English, this basically just means that instead of directly describing the value of a function, we instead describe the function as a bunch of waves that add up to that value. For example, instead of describing a sound in terms of its pressure at any given moment at each point in space, we might describe it as a sum of various sound-waves with their own frequencies and amplitudes.

As for the connection to quantum-mechanics, it'd all go back to the double-slit experiment:

The experiment belongs to a general class of "double path" experiments, in which a wave is split into two separate waves that later combine into a single wave. Changes in the path lengths of both waves result in a phase shift, creating an interference pattern.

—"Double-slit experiment", Wikipedia

One may've expected light-sources to add up, sorta like how one might assume that adding more sound always creates a louder noise (rather than noise-cancellation), but historically they were surprised by the interference pattern that heavily implicated wave mechanics, with a connection to the concept of phase as in the quote above.

In short..

When combined, waves can add or cancel. They add when they largely share the same phase, while they can cancel if they're anti-phased.

So "phase" describes how a wave'll interact with others when combined.

The OP clarified:

It's been a while since I've read QED, but I do recall discussion of 'little clocks'. I'm just having trouble envisioning what those little clocks actually are. I have seen phase as color visualizations. They're very cool to watch, but I don't get what's physically changing in the wavefunction, if that makes sense.

I liked Feynman's way of explaining that. I don't have the book here but as I remember he explains that the dials are just a way to describe they behavior of particles in the quantum world. There is no way to say what those little clocks "actually are". Feynman deliberately chose a very unphysical representation. In the video that the OP linked to these arrows are turning. It is difficult to make such movies, and I find it hard to watch them.

In the same way one cannot say what is "physically changing" in the color visualizations. It is much easier to make such movies and they can also help create intuitions about the behavior of wave functions.

Both these visualisations represent the phase factor $e^{iEt/hbar}$ but that is also just a mathematical description of what is observed: diffraction and interference. The empirical observation is that particles behave like waves at the quantum scale, can interfere destructively. We are not used to that in our everyday experience. That is why quantum mechanics is weird.

Consider Zeno's paradoxes in light of the Einstein's thought experiment of the two lightnings hitting a train. The "paradoxes" show that we can't really talk about objects outside the context of observation and the problems that arise if we postulate existence of object outside of an observer's frame of reference and thus not subjected to the mechanics of perception.

To understand perceived objects we must understand how perception works because anything we think about as a real object is included in the scope of perception, is perceived and is absolutely subjected to it's mechanics.

Zeno's paradoxes are given in terms of what is seen or otherwise perceived, such as a race seen by the eye. When we observe a race where some fast moving object passes by [overtakes] a slower moving object at some point we can't tell who is ahead but after that point the faster moving object is perceived to be ahead, measurement is a matter of precision but is likewise an act of observation.

As I understand it, the wave function is a predictive model describing a to-be-perceived collapse of the function rather than an object's non perceived state.

The short answer is no. No one can give a physical intuition for the meaning of phase, because it has no physical meaning. To clarify this, it may help to recognise that the wave function of a plane wave state in quantum mechanics is not simply a sin wave. It is a helix in the complex plane.

When the helix rotates (as it does under time evolution), the apparent waves on the real and imaginary parts appear to move like waves. But the rotation does not take place in physical space. It takes place only mathematically in a complex configuration space, and the absolute angular position of the helix has no physical meaning.

Wave functions are better called probability amplitudes. They are not real, but are part of the calculation of probabilities for measurement results. The reason we need them is buried deep in the mathematical foundations of quantum mechanics and is not covered in undergraduate (or even most graduate level) courses of quantum theory, which are concerned with applications not conceptual foundations. They are solutions of the Schrodinger equation, and it can be shown that the general form of the the Schrodinger equation is required in a probability theory for indeterminate processes, as distinct from classical probability theory in which outcomes are determined by unknown variables.

(figure from The Mathematics of Gravity and Quanta)

The short answer is that nobody really knows. To understand what the phase physically means would required a detailed physical understanding of the wave function, that is, of quantum mechanics. Alas, it is not known why point particles such as electrons are described, in a probabilistic manner, by - complex - wave functions. We only understand the phase in terms of the wave function. If that is enough for you then the answers above should be sufficient.